Question: The matrix $\mathbf{A} = \begin{pmatrix} 2 & 3 \\ 5 & d \end{pmatrix}$ satisfies
\[\mathbf{A}^{-1} = k \mathbf{A}\]for some constant $k.$  Enter the ordered pair $(d,k).$
Answer: For $\mathbf{A} = \begin{pmatrix} 2 & 3 \\ 5 & d \end{pmatrix},$
\[\mathbf{A}^{-1} = \frac{1}{2d - 15} \begin{pmatrix} d & -3 \\ -5 & 2 \end{pmatrix}\]Comparing entries to $k \mathbf{A},$ we get
\begin{align*}
\frac{d}{2d - 15} &= 2k, \\
\frac{-3}{2d - 15} &= 3k, \\
\frac{-5}{2d - 15} &= 5k, \\
\frac{2}{2d - 15} &= dk.
\end{align*}If $k = 0,$ then $\mathbf{A}^{-1} = \mathbf{0},$ which is not possible, so $k \neq 0.$  Thus, we can divide the equations $\frac{d}{2d - 15} = 2k$ and $\frac{-3}{2d - 15} = 3k$ to get
\[\frac{d}{-3} = \frac{2}{3}.\]Then $d = -2.$  Substituting into the first equation, we get
\[2k = \frac{-2}{2(-2) - 15} = \frac{2}{19},\]so $k = \frac{1}{19}.$  Thus, $(d,k) = \boxed{\left( -2, \frac{1}{19} \right)}.$